14.3 Higher-dimensional and ∞-topoi

Higher-dimensional topoi take ordinary topoi to new heights, incorporating higher categorical structures. They model complex algebraic and geometric relationships, providing a framework for studying advanced mathematical concepts.

∞-topoi push this even further, allowing for infinite hierarchies of morphisms. They capture rich homotopical information and play a crucial role in unifying homotopy theory with category theory, opening doors to new mathematical frontiers.

Higher-Dimensional Topoi

Concept of higher-dimensional topoi

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• Higher-dimensional topoi generalize ordinary topoi to higher categorical settings incorporating higher-dimensional categorical structures (2-categories, n-categories)
• n-topoi model (n-1)-types in homotopy theory capturing higher-dimensional algebraic and geometric relationships
• Key features include internal logic allowing formal reasoning within the topos and geometric realization connecting abstract structures to concrete spaces
• Role in higher category theory provides framework for studying higher categorical structures (higher groupoids, higher stacks)

Theory of ∞-topoi

• ∞-topoi generalize ordinary topoi to infinite-dimensional settings based on ∞-categories allowing infinite hierarchies of morphisms
• Fundamental properties include descent theory for gluing local data and sheaf condition in ∞-categorical context enabling consistent information assembly
• Comparison with ordinary topoi reveals richer structure capturing homotopical information (weak equivalences, higher homotopies)
• Examples of ∞-topoi include ∞-category of spaces and ∞-category of ∞-groupoids representing fundamental objects of study

Connections and Applications

Topoi in homotopy theory

• Higher topos theory unifies concepts from homotopy theory and category theory providing framework for abstract homotopy theory (homotopy types, homotopy groups)
• Homotopy type theory connects to higher-dimensional topoi through univalent foundations program formalizing mathematics in type theory
• Model categories relate to higher-dimensional topoi through Quillen model structures enabling rigorous study of homotopy theory
• Homotopy limits and colimits interpret in higher-dimensional topoi as universal constructions preserving homotopical information
• Grothendieck ∞-groupoids serve as fundamental objects in higher topos theory representing higher-dimensional analogues of sets

Applications of higher-dimensional topoi

• Derived algebraic geometry uses ∞-topoi as foundation for studying derived schemes and stacks generalizing classical algebraic geometry
• Higher-dimensional moduli spaces employ moduli stacks as objects in ∞-topoi enabling study of deformation theory in higher categorical settings
• Brave new algebra utilizes spectral schemes and structured ring spectra extending algebraic concepts to homotopical settings
• Higher categorical cohomology theories generalize cohomology theories in ∞-topoi allowing refined invariants for geometric objects
• Applications in physics include higher gauge theory and topological quantum field theories modeling higher-dimensional phenomena